Multiple equilibria in a monopoly market
with heterogeneous agents and
externalities
Jean-Pierre Nadal (1), Denis Phan (2,3)
Mirta B. Gordon(3) and Jean Vannimenus (1)
(1) Laboratoire de Physique Statistique, Ecole Normale Supérieure,
24 rue Lhomond, 75231 Paris cedex 05, France
([email protected], http//www.lps.ens.fr/˜nadal)
(2) ENST Bretagne, Economic Department, and ICI (Information, Coordination, Incitation), Université de
Bretagne Occidentale, Faculté de Droit, Sciences Economiques et Gestion, Brest, France
([email protected], http//www-eco.enst-bretagne.fr/˜phan)
(3) Laboratoire Leibniz-IMAG,
46, Ave. Félix Viallet, 38031 Grenoble Cedex 1, France
([email protected], web: http//www-leibniz.imag.fr/Apprentissage)
Short title: Multiple equilibria in a monopoly market
IOP PACS/Classification: 89.65.Gh - A-B MOD - BOUND RAT - MARK D&P
Abstract
We explore the effects of social influence in a simple market model in which a large
number of agents face a binary choice: to buy/not to buy a single unit of a product at a price
posted by a single seller (monopoly market). We consider the case of positive externalities: an
agent is more willing to buy if other agents make the same decision. We consider two special
cases of heterogeneity in the individuals’ decision rules, corresponding in the literature to
the Random Utility Models of Thurstone, and of McFadden and Manski. In the first one
the heterogeneity fluctuates with time, leading to a standard model in Physics: the Ising
model at finite temperature (known as annealed disorder) in a uniform external field. In
the second approach the heterogeneity among agents is fixed; in Physics this is a particular
case of quenched disorder models known as random field Ising model, at zero temperature.
We study analytically the equilibrium properties of the market in the limiting case where
each agent is influenced by all the others (the mean field limit), and we illustrate some
dynamic properties of these models making use of numerical simulations in an Agent based
Computational Economics approach.
Considering the optimisation of the profit by the seller within the case of fixed heterogeneity with global externality, we exhibit a new regime where, if the mean willingness to pay
increases and/or the production costs decrease, the seller’s optimal strategy jumps from a
solution with a high price and a small number of buyers, to another one with a low price and
a large number of buyers. This regime, usually modelled with ad-hoc bimodal distributions of
the idiosyncratic heterogeneity, arises here for general monomodal distributions if the social
influence is strong enough.
1
Introduction
Following Kirman [9, 10], we view a market as a complex interactive system with a
communication network. We consider a market with discrete choices [1], and explore
the effects of localised externalities (social influence) on its properties. We focus on
a simple case: a single homogeneous product sold by a single seller (monopoly), and
a large number of heterogeneous customers. These customers are assumed to be
myopic. The only cognitive agent in the model is the monopolist, who determines
the price in order to optimise his profit.
Agents are assumed to have idiosyncratic willingness-to-pay (IWP) described
by means of random variables, as suggested by Thurstone [20] and McFadden [12],
leading to the so-called Random Utility Models [13] (hereafter referred to as RUM).
We first compare and contrast two models: on one hand, the IWP are randomly
chosen and remain fixed, on the other hand, the IWP present independent temporal
fluctuations around a fixed (homogeneous) value. In the present paper we show that
they correspond to two different statistical physics models, in the former case with
quenched disorder (QRUM), and in the latter - the Thurstone case - with annealed
disorder (TRUM). In both cases we assume that the heterogeneous preferences of the
agents are drawn from the same (logistic) distribution. More generally, these models
may be considered as the extreme limits of a single model where the IWPs are time
dependent variables. If the time scale of the changes in the IWP is slow enough to
allow myopic agents to reach the equilibrium upon repeated choices before the IWPs
change, we are in the quenched disorder case (QRUM), whereas if the IWPs vary
at the same pace as the individual decisions, we are in the annealed limit (TRUM).
The equilibrium states of the two models generally differ, except in the special case
of homogeneous interactions with complete connectivity. In this special situation,
which corresponds to a mean-field model in physics, the expected aggregate steadystate is the same in both models.
Considering the optimisation of the profit by the seller in the quenched disorder case (QRUM), we exhibit a new regime where, if the mean willingness to pay
increases and/or the production costs decrease, the seller’s optimal strategy jumps
from a solution with a high price and a small number of buyers, to another one
with a low price and a large number of buyers. To our knowledge, this transition in
the monopolist’s startegy, which has the characteristics of what is called a first order phase transition in Physics, is usually modelled by assuming an ad-hoc bimodal
heterogeneity in the agents’IWPs. We find that it exists even with a monomodal
distribution, if the social influence is strong enough. We find also that it occurs not
only in the domain of parameters where the demand itself, at a given price, presents
two solutions, but also and more surprisingly, in a domain where the demand is
uniquely defined.
In the following, we first present the demand side, and then consider the optimisation problem left to the monopolist, who is assumed to know the demand
model and the distribution of the IWP over the population, but cannot observe the
individual (private) values.
2
Simple models of discrete choice with social influence
We consider a set ΩN of N agents with a classical linear IWP function. Each agent
i ∈ ΩN either buys (ωi = 1) or does not buy (ωi = 0) one unit of a homogeneous
good sold by a unique seller. A rational agent chooses ω i in order to maximise his
2
surplus function Vi :
max Vi = max ωi (Hi +
ωi ∈{0,1}
ωi ∈{0,1}
X
k∈ϑi
Jik ωk − P ),
(1)
where P is the price of one unit and Hi represents the idiosyncratic preference
component: in the absence of social influence, H i is the reservation price of agent
i, i.e., the maximum price he is willing to pay for the good. In addition, in (1) we
assume that the preferences of each agent i are influenced by the choices of a subset
ϑi ⊆ ΩN of other agents, hereafter called neighbours of i. This social influence is
represented by a weighted sum of these choices, ω k with k ∈ ϑi . The corresponding
weight, denoted by Jik , is the marginal social influence of the decision of agent k ∈ ϑ i
on agent i. When this social influence is assumed to be positive (J ik ≥ 0), its effect
may be identified, following Durlauf [4], as a strategic complementarity in agents’
choices [3].
For simplicity we consider here only the case of homogeneous influences, that is
identical neighbourhood structures ϑ of cardinal n for all the agents, and identical
positive weights, that we write Jik = J/n. That is,
∀i ∈ ΩN : |ϑi | = n, and: ∀k ∈ ϑi Jik = J/n > 0.
(2)
Hence for a given distribution of choices in the neighbourhood ϑ i , and for a given
price, the customer’s behaviour is deterministic: an agent buys if
Hi > P −
2.1
J X
ωk .
n k∈ϑ
(3)
i
Psychological versus economic points of view
At the basis of Ramdom Utility Models (RUM) [11, 13], the discrete choice model
(1) may represent two quite different situations, depending on the nature of the
idiosyncratic term Hi . Following the typology proposed by Anderson et al. [1], we
distinguish a “psychological” and an “economic” approach to individual choices.
Within the psychological perspective of Thurstone [20], the utility has a stochastic
aspect because “there are some qualitative fluctuations from one occasion to the
next... for a given stimulus”. Hereafter we refer to this model as the Thurstone
Random Utility Model (TRUM). On the contrary, for McFadden [12] and Manski [11],
each agent has a willingness to pay that is invariable in time, but may differ from
one agent to the other. We call this perspective the QRUM case, where ’Q’ stands
for Quenched for reason explicited in the next section. Accordingly, the TRUM and
the QRUM perspectives differ in the nature of the individual willingness to pay.
We assume that the seller is an external observer in a risky situation: he cannot
observe the individual values of the IWPs. He considers that the heterogeneous set
of N IWPs are a sample of N i.i.d. random variables, and we assume that he knows
its statistical distribution over the population.
In the TRUM, the idiosyncratic preferences H i (t) are time-dependent i.i.d. random variables. The agents are identical in that the H i (t) are drawn at each time t
3
from a same probability distribution, which we characterise by its mean H and the
cumulative distribution F (z) of the deviations from the mean,
F (z) ≡ P(Hi − H ≤ z).
(4)
In the case of a logistic distribution with mean H, and variance σ 2 = π 2 /(3β 2 )
(where H and β are thus constant, independent of both i and t), the cumulative
distribution F (z) is:
1
.
(5)
F (z) =
1 + exp(−β z)
In the QRUM, agents are heterogeneous: the private idiosyncratic terms H i
are randomly distributed over the agents, but remain fixed during the period under
consideration. If we assume that the H i are logistically distributed with mean H and
variance σ 2 = π 2 /(3β 2 ), then at a given instant of time t, the empirical distribution
of the IWP in the population is the same in the TRUM and the QRUM approaches:
in the large N limit they are given by the same logistic distribution, with the same
mean and same variance.
2.2
Static versus dynamic points of view
If the agents make repeated choices, in the standard TRUM the H i are newly drawn
from the logistic distribution, whereas in the QRUM the H i remain the same.
The analysis of the dynamical evolution of the models in statistical physics,
corresponds to the case of repeated choices and myopic agents. These apply at
each time step the decision rule (3), based on the observations of the choices at the
previous time. Under these hypothesis, an equilibrium or steady state of the overall
system is reached. It may differ in both models, as shortly explained below.
2.3
“Annealed” versus “quenched” disorder
Since we have assumed isotropic (hence symmetric) interactions, there is a strong
relation between these models and Ising type models in Statistical Mechanics, which
is made explicit if we change the variables ω i ∈ {0, 1} into spin variables si ∈ {±1}
through si = 2ωi − 1. All the expressions in the present paper can be put in terms
of either si or ωi using this transformation. In the following we keep the encoding
ωi ∈ {0, 1}. The economics assumption of strategic complementarity corresponds to
having ferromagnetic couplings in physics (that is, the interaction J between Ising
spins is positive). These couplings act as a bias favoring states where the spins s i
are aligned with each other, that is, they tend to take all the same value.
The TRUM corresponds to a case of annealed disorder. Having a time varying
random idosyncratic component is equivalent to introducing a stochastic dynamics
for the Ising spins. In the particular case where F (z) is the logistic distribution,
we obtain an Ising model in a uniform (non random) external field H − P , at
temperature T = 1/β. Although the individual choices change at each time setp
due to the randomness in the Hi (t), the aggregate fraction of consumers in the
long run fluctuates gently around a well defined stationary value. The QRUM has
fixed heterogeneity; it is analogous to a Random Field Ising Model (RFIM) at zero
4
temperature, that is, with deterministic dynamics. The RFIM belongs to the class of
quenched disorder models: the values H i are equivalent to random time-independent
local fields. The “agreement” among agents favored by the ferromagnetic couplings
may be broken by the influence of these heterogeneous external fields H i . Due to the
random distribution of Hi over the network of agents, the resulting organisation may
be complex. Thus, from the physicist’s point of view, the TRUM and the QRUM are
quite different models: uniform field and finite temperature in the former, random
field and zero temperature in the latter.
The properties of disordered systems have been and still are the subject of numerous studies in statistical physics. They show that annealed and quenched disorder
can lead to very different behaviours. The standard Ising model (the TRUM case)
is well understood. In the case where the agents are situated on the vertices of a 2dimensional square lattice, and have four neighbours each, there is an exact analysis
of the stationary states of the model for P = P n (the neutral case, see below equation (35)) due to Onsager [14]. Even if an analytical solution of the optimization
problem (1) for an arbitrary neighbourhood does not exist, the mean field analysis
gives approximate results that become exact in the limiting situation where every
agent is a neighbour of every other agent (i.e. all the agents are interconnected
through weights (2)). On the contrary, the properties of the RFIM (the QRUM case
with externalities) are not yet fully understood. Since the first studies of the RFIM,
which date back to Aharony and Galam [6, 5], a number of important results have
been published in the physics literature (see e. g. [18]). Several variants of the RFIM
have already been used in the context of socio-economic modelling [7, 15, 2, 21].
2.4
Mean-field version of the Random Utility Model with externalities
Hereafter, we restrict our investigation to the QRUM in the case of a global externality. That is, we consider homogeneous interactions and full connectivity, which
is equivalent to considering the mean field version of the RFIM in physics. Within
this general framework, we are interested in two different perspectives. First we
consider a static point of view: by looking for the equality between demand and
supply we determine the set of possible economic equilibria (the Nash equilibria).
This will allow us to analyse in section 4 the optimal strategy of the monopolist,
as a function of the model parameters. Then (section 5) we consider the market’s
dynamics assuming myopic agents: based on the observation of the behaviour of the
other agents at time t − 1, each agent decides at time t to buy or not to buy (this
corresponds to a myopic best-reply strategy). We show that, in general, the market
converges towards the static equilibria, except for a precise range of the parameter
values where interesting static as well as dynamic features are observed.
These two kinds of analysis correspond in Physics to the study of the thermal
equilibrium properties within the statistical ensemble framework on the one hand,
and the out of equilibrium dynamics (which, in most cases, approaches the static
equilibrium through a relaxation process) on the other hand.
5
3
Aggregate demand
For the QRUM, it is convenient to decompose H i as
Hi = H + θ i ,
(6)
where H is the mean value of the Hi ’s over the population, and θi the idiosyncratic
component which characterises how much the IWP of agent i deviates from the
mean. We assume the θi to be i.i.d. random variables of zero mean and variance
σ 2 = π 2 /(3β 2 ), with a logistic distribution so that their cumulative distribution
F (z) = P(θi ≤ z) is given by (5).
With (6) we can then rewrite the decision rule (3) as
θi > P − H −
J X
ωk
n k∈ϑ
(7)
i
As discussed in the preceding section, we consider the full connectivity case with
isotropic interactions given by (2) with cardinal n = N − 1, in the limit of a very
large number of agents. Under these conditions the social influence term of the
agents’ surplus function (the coefficient of J in the above equation (7)), equal to
P
k∈ϑ ωk /(N − 1), can be approximated by the penetration rate η, defined as the
fraction of customers that choose to buy at a given price:
η ≡ lim
N →∞
N
X
ωk /N.
(8)
k=1
The condition of buying, given by equation (7) may thus be replaced by
ωi = 1 iff θi > z,
(9)
z ≡ P − H − J η.
(10)
where z is defined by
Equations (9) and (10) allow us to obtain η as a fixed point:
η = 1 − F (z)
(11)
where z depends on P , H, and η, and F (z) is the cumulative distribution of the
IWP around the average value H. Note that this (macroscopic) equation is formally
equivalent to the (microscopic) individual expectation that ω i = 1 in the TRUM
case. Using the logistic distribution for θ i , we have:
η=
1
1 + exp (+ β z)
(12)
Equation (11) allows us to define η as an implicit function of the price through
Φ(η, P ) ≡ η(P ) + F (P − H − J η(P )) − 1 = 0.
6
(13)
Since for a given P , equation (12) defines the penetration rate η as a fixed-point,
inversion of this equation gives an inverse demand function:
P d (η) = H + J η +
1 1−η
ln
β
η
(14)
At given values of β, J and H, for most values of P , (12) has a unique solution
η(P ). However for βJ > βJB ≡ 4, there is a range of prices
P1 (βJ, βH) < P < P2 (βJ, βH)
(15)
such that, for any P in this interval, (12) has two stable solutions and an unstable
one. The limiting values P1 and P2 are the particular price values obtained from
the condition that eq. (12) has one degenerate solution:
η = 1 − F (z), and
d(1 − F (z))
= 1.
dη
The second equation gives βJη(1 − η) = 1, which has two solutions, η 2 ≤ 1/2 ≤ η1 ,
"
1
1±
ηi =
2
s
4
1−
βJ
#
; i ∈ {1, 2}
(16)
Note that ηi depends only on βJ. Then, the limiting prices P i are equal to the
inverse demands associated with these values η i , which are, from (14):
Pi = H + Jηi +
1
1 − ηi
] ; i ∈ {1, 2}.
ln[
β
ηi
(17)
Note that these limiting prices are not necessarily positive.
It is interesting to note that the set of equilibria is the same as what would be
obtained if agents had rational expectations about the choices of the others: if every
agent had knowledge of the distribution of the H i , he could compute the equilibrium
state compatible with the maximisation of his own surplus, taking into account that
every agent does the same, and make his decision (to buy/not to buy) accordingly.
For βJ < βJB every agent could thus anticipate the value of η to be realized at
the price P , and make his choice according to (9). For βJ > βJ B , however, if the
price is set within the interval [P1 , P2 ], the agents are unable to anticipate which
equilibrium will be realized, even though the one with the largest value of η should
be preferred by every one (it is the Pareto dominant equilibrium). Remark that
such a situation is similar to the one arising in stag hunt-type coordination games.
4
Supply side
On the supply side, we consider a monopolist facing heterogeneous customers in a
risky situation where he has perfect knowledge of the functional form of the agents’
surplus functions and their maximisation behaviour (1). He also knows the statistical
(logistic) distribution of the idiosyncratic reservation prices (H i ), but cannot observe
any individual reservation price. He observes only the aggregated individual choices
7
(which are either to buy or not to buy). The social influence on each individual
decision is then close to J η, and the fraction of customers η is observed by the
monopolist. That is, for a given price, the expected number of buyers is given by
equation (11).
Remark that, although we have restricted our analysis to the QRUM case, the
probability for an agent taken at random by the monopolist to be a customer would
be formally the same in the TRUM case, as already pointed out by McFadden [13].
4.1
Profit maximisation
Let C be the monopolist cost for each unit sold, so that
p≡P −C
(18)
is his profit per unit. Since P − H = (P − C) − (H − C), defining
h ≡ H − C,
(19)
z = p − h − J η.
(20)
we can rewrite z in (10) as:
Hereafter we write all the equations in terms of p and h (hence we will also make
use of pd (η) ≡ P d (η) − C).
Since each customer buys a single unit of the good, the monopolist’s total expected profit is p N η, which is proportional to the total number of customers. He
is left with the following maximisation problem:
pM = arg max Π(p),
p
(21)
where N Π(p) is the expected profit, with:
Π(p) ≡ p η(p),
(22)
and where η(p) is the solution to the implicit equation (13). If there is no discontinuity in the demand curve η(p) (hence for βJ ≤ 4), p M satisfies dΠ(p)/dp = 0,
which gives dη/dp = −η/p at p = pM . Using the implicit derivative theorem, from
(13) one has:
−∂Φ/∂P
−f (z)
dη(P )
=
=
(23)
dP
∂Φ/∂η
1 − J f (z)
where f (z) = dF (z)/dz is the probability density. Thus we obtain at p = p M :
f (z)
η
= ,
1 − Jf (z)
p
(24)
where z, defined in (20), has to be taken at p = p M .
Because the monopolist observes the demand level η, we can use equation (11)
to replace 1 − F (z) by η. Making use of the explicit form of F (z) (the logistic (5)),
8
one has f (z) = βF (z)(1 − F (z)) = β(1 − η)η. Then equation (24) can be written as
p = ps (η) where the function ps (η) is defined by:
1
−J η
β (1 − η)
(25)
pM = pd (ηM ) = ps (ηM ),
(27)
ps (η) ≡
A more general and formal analysis shows that, for an arbitrary function F , p s is
given by:
dpd (η)
ps (η) ≡ −η
,
(26)
dη
and the interesting structure of the optimisation program gives to this function
ps (η) the role of an effective (inverse) supply function [22]. Since it is not a true
supply function, but results from the monopolist’s optimisation program based on
the knowledge of the demand function, we will refer to p s (η) as the implied inverse
supply function. We thus obtain pM and ηM as the intersection between demand
(14) and (implied) supply (25):
where pd = P d − C.
The (possibly local) maxima of the profit are the solutions of (27) for which
d2 Π
< 0.
dp2
(28)
It is straightforward to get the expression for the second derivative of the profit:
η
2η − 1
d2 Π
= −2 1 +
,
2
dp
p
2βp(1 − η)2
(29)
from which it is clear that the solutions with η > 1/2 are local maxima. For η < 1/2,
condition (28) reads
1 − 2η
< 1.
(30)
2βp(1 − η)2
Making use of the above equations, this can also be rewritten as
2βJη(1 − η)2 < 1.
(31)
For βJ > βJB = 4, the monopolist has to find p = pM which realises the
programme:
M
pM : max{Π− (pM
(32)
− ), Π+ (p+ )}
pM
+ = arg max Π+ (p) ≡ p η+ (p),
(33)
pM
− = arg max Π− (p) ≡ p η− (p)
(34)
p
p
where the subscripts + and − refer to the solutions of (12) with a fraction of buyers
larger, respectively smaller, than 1/2.
To illustrate the behaviour of these equations, figure 1 represents several examples of inverse supply and demand curves corresponding to different market configurations.
9
Figure 1:
Inverse demand and implied supply curves pd (η) and ps (η), for different values of h and J
(β = 1, C = 0 hence h = H). The equilibrium prices are obtained at the intersection between the demand
(black) and the supply (grey) curves.
The two graphics on the top illustrate the difference between a complete absence of externality (J = 0,
dashed lines) and a strong externality (J = 4, solid lines). The case h = 2 (left) corresponds to a strong
positive average of the population’s IWP (h = 2), whereas the population is neutral for h = 0 (right).
The values of h and J in the four graphics labelled (a) to (d) correspond to the points (a) to (d) in the
phase diagram (figure 3). They all have negative values of the average of the population’s IWP (h < 0), so
that in the absence of externality only few consumers would be interested in the single commodity.
(a) corresponds to the coexistence region between two local market equilibria in figure 3; but one of them
(not shown) is not relevant since it corresponds to a negative price solution. (b) lies also in the coexistence
region; in this case, the optimal market equilibrium is the one with high η. (c) lies in the region with only
one market equilibrium, with few buyers (small η). (d) corresponds to a large social effect; the single market
equilibrium has large η and a high price.
10
0.8
β h=-2
η_
0.6
β pM_
βΠ M_
0.4
η+
β pM+
0.2
βΠ M+
0.0
3.4
3.6
3.8
4.0
4.2
4.4
4.6
βJ
Figure 2: Fraction of buyers η, optimal price βpM and monopolist profit ΠM , as a function of the social
influence, for βh = −2. The superscripts − and + refer to the two solutions of equations (27) that are
relative maxima.
-1.0
phase transition line
(d)
A
-1.5
-2.0
B
βh
-2.5
-3.0
(a)
(b)
(c)
3
4
5
βJ
6
7
Figure 3: Phase diagram in the plane {βJ, βh}: the grey region represents the domain in the parameter
space where coexist two maxima of the monopolist’s profit, a global one (the optimal solution) and a local
one. Inside this domain, as βJ and/or βh increase, there is a (first order) transition where the monopolist’s
optimum jumps from a high price, low penetration rate solution η = η− to one with low price, large η = η+ .
The circles on the transition line have been obtained numerically, the smooth curves are obtained analytically
(see the Appendix and [22] for details). The points (a) to (d) correspond to the inverse supply and demand
curves represented in figure 1. In the white region, for βJ < 27/8, the fraction of buyers, η, increases
continuously from 0 to 1 as βh increases from −∞ to +∞ (c-d). At the singular point A, (βJ = 27/8,
βh = −3/4 − log(2)), η+ = η− = 1/3. At point B (βJ = 4, βh = −2), the local maximum with η large
appears with a null profit and η+ = 1/2. In the dark-grey region below B, this local maximum exists with
a negative profit, being thus non viable for the monopolist (a).
11
4.2
Phase transition in the monopolist’s strategy
In this section we analyse and discuss the solution of the optimal supply-demand
static equilibria, that is, the solutions of equations (27) and (31). As might be
expected, the result for the product ηp M depends only on the two parameters βh
and βJ. That is, the variance of the idiosyncratic part of the reservation prices fixes
the scale of the important parameters, and in particular that of the optimal price.
This is why we present our results under the form of a phase diagram with axis
(βJ, βh), as usual in physics (see figure 3). Each point in this diagram corresponds
to a particular set of parameters of the customers-monopolist system. The lines
represent boundaries between regions of qualitatively different equilibria, that we
describe hereafter.
Let us first discuss the case where h > 0. It is straightforward to check that in
this case there is a single solution η M . It is interesting to compare the value of p M
with the value pn corresponding to the neutral situation on the demand side. The
latter corresponds to the unbiased situation where, on average, there are as many
agents likely to buy as not to buy (η = 1/2). Since the expected willingness to pay
of any agent i is h + θi + J/2 − p, its average over the set of agents is h + J/2 − p.
Thus, the neutral state is obtained for
pn = h + J/2.
(35)
To compare pM with pn , it is convenient to rewrite equation (14) as
β(pd − pn ) = βJ(η − 1/2) + ln[(1 − η)/η].
(36)
This equation gives pd = pn for η = 0.5, as it should. For this value of η, equation
(25) gives ps = pn only if β(h + J) = 2: for these values of J and h, the monopolist
maximises his profit when the buyers represent half of the population. When β(h +
J) increases above 2 (decreases below 2), the monopolist’s optimal price decreases
(increases) and the corresponding fraction of buyers increases (decreases).
Finally, if there are no social effects (J = 0) the monopolist optimal price is a
solution of the implicit equation:
pM =
1 + exp (−β(pM − h))
1
=
.
β F (pM − h)
β
(37)
The value of βpM lies between 1 and 1 + exp(βh). Increasing β lowers the optimal
price: since the variance of the distribution of willingness to pay gets smaller, the
only way to keep a sufficient number of buyers is to lower the prices.
Consider now the case with h < 0, that is, on average the population is not willing
to buy. Due to the randomness of the individual’s reservation price, H i = H + θi ,
the surplus may be positive but only for a small fraction of the population. Thus,
we would expect that the monopolist will maximise his profit by adjusting the price
to the preferences of this minority. However, if the social influence represented by
J is strong enough, this intuitive conclusion is not supported by the solution to
equations (27). The optimal monopolist’s strategy shifts abruptly from a regime
of high price and a small fraction of buyers to a regime of low price with a large
12
fraction of buyers as βJ increases. Such a discontinuity might actually be expected
for βJ > 4 = βJB , that is when the demand itself has a discontinuity. But, quite
interestingly, the transition is also found in the range βJ A ≡ 27/8 < βJ < 4 = βJB ,
that is, in a domain of the parameters space (βJ, βh) where the demand η(p) is a
smooth function of the price.
Such a transition is analogous to what is called a first order phase transition
in physics [19]: at a critical value βJ c (βh) of the control parameter the fraction
of buyers jumps from a low to a high value. Before the transition, above a value
βJ− (βh) < βJc (βh) equations (27) already present several solutions. Two of them
are local maxima of the monopolist’s profit function, and one corresponds to a local
minimum. The global maximum is the solution corresponding to a high price with
few buyers for βJ < βJc , and that of low price with many buyers for βJ > βJ c .
Figure 2 presents these results for the particular value βh = −2, for which it can be
shown analytically that βJ− = 4, and βJc ≈ 4.17 (determined numerically).
The detailed discussion of the full phase diagram in the plane (βJ, βh), shown
on Figure 3, is presented in the Appendix, and a more general discussion will be
presented elsewhere [22].
5
Dynamic features
In RUM models, the individual thresholds of adoption implicitly embody the number
of agents each individual considers sufficient to modify his behaviour, as underlined
in the field of social science [17, 8]. We briefly discuss here some dynamical aspects
of the QRUM, considering a market with myopic customers: each agent makes its
decision at time t based on the observation of the behaviour of the other agents at
time t−1, that is, the agents have a myopic best-reply strategy. The adoption by very
few agents in the population (the “direct adopters”) may then lead to a significant
change in the whole population through a chain reaction of “indirect adopters” [16].
This chain reaction depends on the type of dynamics considered, synchronous (all
the agents take their decision at the same time, based on the previous decisions
of their neighbours) or asynchronous (at each time step a single agent, picked at
random, makes his decision). In synchronous dynamics, the fraction of adopters in
the large N limit is then given by
η(t) = 1 − F (P − H − J η(t − 1))
(38)
and η(t) converges to a solution of the fixed point equation (11). As we have seen,
there are values of J, H and P (those of the shaded region in the phase diagram of
Figure 3) for which (11) presents two stable and one unstable fixed points provided
that β is large enough (small σ). At a given price, the stable solutions correspond
to two possible levels of η (Figure 4a).
Consider the following monopolist strategy: start with a price sufficiently low (or
sufficiently high) to be in the region where only one solution exists for the fraction
of buyers. Then, by increasing (decreasing) the price smoothly, the equilibrium
fraction of buyers at each price converges to the corresponding fixed point solution.
At some price the system jumps abruptly to the other fixed point solution. However,
13
if the price is decreased (increased) back, the jump occurs at a different price. This
phenomenon, called hysteresis, is characteristic of the so called first order phase
transitions in Physics. At such transitions, some extensive property of the system
(here, the fraction of buyers) changes abruptly when the parameters (here, the price)
are infinitesimally modified.
In the region of the phase diagram where two solutions exist, the monopolist
is not guaranteed that the fraction of customers will be the fraction expected by
his profit optimization program. Generally, when the price is slightly changed, the
number of customers between two fixed point solutions evolves through a series
of clustered flips (between ωi = 1 and ωi = 0). The resulting global change is
referred to as an avalanche. Notice that several successive updates of the customers’
decisions are needed to reach the corresponding fixed point solution. In the case of
bounded agents’ neighbourhoods (not discussed in this paper), when the system is at
one equilibrium point on the hysteresis cycle, secondary inner hysteresis loops, that
start and end at the same point, may be produced by changing back and forth the
price. This complex behaviour of RFIMs, first described by Sethna [18], has been
discussed within the economics context of QRUM with externalities by Pajot et
al [16]. Note that these secondary hysteresis phenomena are specific to the QRUM;
they are not present in the TRUM.
In the present case of a global neighborhood, there is a single huge avalanche
involving all the agents that change their states, at the corresponding transition.
Figure 4 illustrates the hysteresis phenomenon. The curves in Figure 4a, represent
the number of customers as a function of the price, obtained through a simulation of
the whole demand system with synchronous dynamics. The black (grey) curve corresponds to the “upstream” (downstream) trajectory, when prices decrease (increase)
in steps of 10−4 , within the interval [0.9, 1.6]. We observe the hysteresis phenomenon
with discontinuous transitions around the theoretical neutral price P n = 1.25, defined by (35). Typically, along the downstream trajectory (with increasing prices,
grey curve) the externality effect induces a strong resistance of the demand system
against a decrease in the number of customers. In both cases, large avalanches occur
at the first order phase transition. Figure 4b represents the sizes of the dramatic
induced effects at the successive updates in the avalanche from one fixed point to
the other, as a function of time (see [16] for more details).
For small enough values of β (large σ), there is always a single fixed point for
all the values of P , and no hysteresis at all. This simpler behaviour is obtained for
parameter values situated in the white region of the phase diagram (Figure 3).
6
Conclusion
In this paper, we have first compared two extreme special cases of discrete choice
models, the Random Utility Model (QRUM) of Manski and McFadden, and the
Thurstone model (TRUM), in which the individuals bear a local positive social
influence on their willingness to pay, and have random heterogeneous idiosyncratic
preferences. In the QRUM the latter remain fixed, and give rise to a complex market
organisation. For physicists, this model with fixed heterogeneity belongs to the class
14
a - Hysteresis in the trade-off between prices
b - Chronology and sizes of induced effects
and customers: upstream (black)
in an avalanche at the phase
and downstream (grey) trajectories
transition for P = 1.2408 (P n = 1.25)
Figure 4:
Number of customers as a function of the price, at the discontinuous phase transition (full
connectivity, synchronous activation regime; source: Phan et al. [16]; parameters: N = 1296, H = 1,
J = 0.5, β = 10).
of “quenched” disorder models; the QRUM is equivalent to a “Random Field Ising
Model” (RFIM). In the TRUM, all the agents share a homogeneous component of the
willingness to pay, but have an additive, time varying, random (logistic) idiosyncratic
characteristic. In physics, this problem corresponds to a case of “annealed” disorder.
The random idiosyncratic component results in a stochastic dynamics, because each
agent decides to buy according to the logit choice function at each time step, making
this model formally equivalent to an Ising model at temperature T 6= 0 in a uniform
(non random) external field. From the physicist’s point of view, the QRUM and
the TRUM are quite different: random field and zero temperature in the QRUM
case, uniform field and non zero temperature in the TRUM case. An important
result in statistical physics is that quenched and annealed disorders can lead to very
different behaviours. In this paper we have briefly discussed some consequences on
the market’s behaviour.
Next, we have considered the QRUM case with a global externality corresponding
to a positive social influence, and with a single seller (the case of a monopoly market).
Studying the optimisation of the profit by the seller, we have exhibited a new “first
order phase transition”: when the social influence is strong enough, there is a regime
where, upon increasing the mean willingness to pay, or decreasing the production
costs, the optimal monopolist’s solution jumps from one with a high price and a
small number of buyers, to one with a low price and a large number of buyers. It is
worth to stress that the multi equilibria domain exists as a consequence of a positive
externality, without assuming any bimodal distribution of the IWPs. Moreover, as we
will show with more details in a forthcoming paper [22], the phase diagram derived
here under the hypothesis of a logit distribution is generic of any smooth monomodal
distribution.
We have only considered fully connected systems: the theoretical analysis of
systems with finite connectivity is more involved, and requires numerical simulations.
The simplest configuration is one where each customer has only two neighbours, one
on each side. The corresponding network, which has the topology of a ring, has been
analysed numerically by Phan et al. [16]) who show that the optimal monopolist’s
price increases both with the degree of the connectivity graph and the range of the
interactions (in particular, in the case of “small world” networks). Buyers’clusters
of different sizes may form, so that it is no longer possible to describe the externality
15
with a single parameter, like in the mean field case. Further studies in computational
economics are required in order to explore such situations.
Acknowledgements
We thank Annick Vignes for a critical reading of the manuscript. This work is part
of the project ’ELICCIR’ supported by the joint program ”Complex Systems in
Human and Social Sciences” of the French Ministry of Research and of the CNRS.
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II, Santa Fe Institute Studies in the Sciences of Complexity, Volume XXVII,
Addison-Wesley, pp. 491-531.
[10] Kirman A.P. (1997b), ”The Economy as an Evolving Network”, Journal of
Evolutionary Economics, 7, pp. 339-353.
16
[11] Manski, C. (1977), ”The Structure of Random Utility Models”, Theory and
Decision, 8, 229-254.
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In Griliches Intrilligator (eds.), Handbook of Econometrics Vol II, Elsevier
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[13] McFadden D. (1976) ”Quantal Choice Analysis: A Survey,” Annals of Economic and Social Measurement, Vol. 5, No. 4, 363-390.
[14] Onsager, L.(1944) ”A 2D model with an order-disorder transition”, Physical
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[15] Orléan A. (1995) ”Bayesian Interactions and Collective Dynamics of Opinion:
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Organization, 28, pp. 257-74.
[16] Phan D., Pajot S., Nadal J.P. (2003) ”The Monopolist’s Market with Discrete
Choices and Network Externality Revisited: Small-Worlds, Phase Transition
and Avalanches in an ACE Framework”. Ninth annual meeting of the Society
of Computational Economics, University of Washington, Seattle, USA, July
11 - 13, 2003.
http://econpapers.hhs.se/paper/scescecf3/150.htm
[17] Schelling T.S. (1978) Micromotives and Macrobehavior, Norton and Co, N.Y.
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[22] A more detailed and general analysis, that is for a large family of IWP distributions, will be published elsewhere (Gordon M. B., Nadal J.-P., Phan D.
and Vannimenus J., in preparation).
17
Appendix: Phase Diagram
In this Appendix we detail the derivation of the phase diagram in the plane (βJ,
βh), presented in figure 3. The phase diagram shows the domain in the parameter
space where coexist two maxima of the monopolist’s profit, one global maximum
(the optimal solution) and one local maximum. Inside this domain there is a (first
order) transition line where, as βJ and/or βh increases, the optimal solution jumps
from a solution ’-’ with a low value η = η − to a solution ’+’ with a large value
η = η+ , η being the fraction of buyers. In figure 3, the circles ’o’ are points on the
transition line obtained numerically; all the other curves being obtained analytically
as explained below.
In the following without loss of generality we set β = 1, which is equivalent to
say that we measure J and h in units of 1/β.
To explain the phase diagram in more detail, it is convenient to parametrise
every quantity/curve as a function of η. First the (per unit) profit p is given by
p=
1
− Jη
1−η
(39)
(hence the profit N Π = N p(η)η), and η is a fixed point of the equation
η = G(h, J, η)
with
G(h, J, η) =
1
1 + exp(−h − 2Jη +
(40)
1
1−η )
(41)
We will also make use of an alternative form of (40), (41), that is
h = −2Jη +
1
η
+ log(
)
1−η
1−η
(42)
One can also show that the fixed point equation (40) is equivalent to
d(ηP d (η))
= 0,
dη
(43)
and that the condition for having a maximum, (29), is equivalent to
d(ηP s (η))
> 0.
dη
(44)
As we will see, there are two singular points of interest:
A: JA = 27/8, hA = −3/4 − log(2);
B: JB = 4,
hB = −2.
Let us describe the phase diagram considering that, at fixed J, one increases h
starting from some low (strongly negative) value.
If J < 27/8 = JA , the optimal solution changes continuously, the fraction of
buyers increasing with no discontinuity from a low to a high value as h increases.
More generally, outside the grey domain in figure 3 there is a unique solution of the
optimisation of the profit.
18
For J > 27/8 = JA , as h increases one will first hit the lower boundary of the
grey region on the phase diagram, h = h − (J). On this line, a local maximum of
the profit appears, corresponding to a value η = η + > 1/3. As shown in figure 5a,
the curve y = G(h, J, η) intersects y = η at some small value η = η − and is tangent
to it at η = η+ . For h− (J) < h < h+ (J) y = G(h, J, η) has three intersects with
the diagonal y = η, h = h+ (J) being the upper boundary of the grey region on
the phase diagram. The stability analysis shows that the two extreme intersects
correspond to maxima of the profit, giving the solutions η = η − and η = η+ . On
the upper boundary h = h+ (J), it is the solution with a small value of η which
disappears, with y = G(h, J, η) becoming tangent to y = η for η = η − , see figures
5c1 and 5c2. These lower and upper boundaries are obtained by writing that the
second derivative of the profit with respect to p is zero, giving
2Jη(1 − η)2 = 1
(45)
Together with (42) this gives the curves parametrised by η,
1
2η(1 − η)2
1
η
1
+
+ log(
)
h = −
2
(1 − η)
1−η
1−η
J
=
(46)
the lower curve h− (J) corresponding to the branch η = η + ∈ [1/3, 1], and the upper
curve h+ (J) corresponding to the branch η = η − ∈ [0, 1/3].
The two curves merge at the singular point A, at which η + = η− = 1/3, JA =
27/8, hA = −3/4 − log(2). Expanding the above equations (46) near η = 1/3 we find
that the two curves are cotangent at A, with a slope −2/3. This common tangent
is thus also tangent to the transition line at A. A straight segment of slope −2/3
starting from A is plotted on the phase diagram, figure 3, and one can see that this
is a very good approximation of the transition line for J < 4 = J B .
On the lower boundary, for J > 4 = JB , the local maximum with η = η+ appears
with a negative profit (zero profit at point B where η + = 1/2). The profit becomes
positive on the curve starting at point B, on which the profit is zero with η + > 1/2.
0 (J),
This curve is obtained by writing p = 0, η + > 1/2, that is η+ = η+
0
η+
(J)
"
1
1+
≡
2
r
4
1−
J
#
(47)
and h is obtained as a function of J, by replacing η in (42) by the above expression
(47). In this domain of negative profit for the local maximum, the distance to the
transition line (at a given value of J) is equal to the amount by which the production
cost per unit of good must be lowered in order to make the solution viable.
In the domain J > 4 = JB , the transition line, computed numerically, appears to
be just above this null profit line. This suggests that an expansion for p small for the
solution η+ , and for η small for the solution η− should provide good approximations.
The transition is obtained when Π+ = Π− as explained below, and this allows to
display the curve of figure 3, which turns out to be a very good approximation of
the transition line for large values of J (or small values of h, typically h < −4.5).
19
Let us first consider the vicinity of the point B at which p + = 0, η+ = 1/2,
and the second derivative of the profit is zero for this ’+’ local solution. Expanding
near J = 4, h just above h− (4) = −2 (p small), one gets the behaviour of the ’+’
solution:
≡ h + 2, 0 < << 1
r
1
(1 +
)
η+ =
2
2
p+ = + o(3/2 )
Π+ =
+ o(3/2 )
2
(48)
The singular, square-root, behaviour of η is specific to point B. For any J > 4,
just above the null curve η+ increases linearly with ≡ h − h0+ (J), where h0+ (J) is
0 (J) defined
the value of h on the null curve (obtained by replacing η in (42) by η +
in (47)). The price and the profit have, however, the same behaviour as for J = 4.
More precisely, at lowest order in , one gets:
0 < = h − h0+ (J) << 1
0
η+ = η +
(J) + p+ = ,
0 (1 − η 0 )2
η+
+
0 (1 − η 0 )2
1 − 2Jη+
+
0
Π+ = η +
(J) .
(49)
One can see from the expression of η+ in (49) how the singularity at point B appears:
the coefficient of diverges when condition (45) is fulfilled, that is when the solution
is marginally stable, which is the case at B.
Similarly one can get the behaviour of the ’+’ solution near point B at h = −2,
increasing J from J = 4, as shown in figure 2
η+ =
p+ =
Π+ =
s
J −4
)
2
√
2
1
(J − 4) +
(J − 4)3/2
2
12
1
1
(J − 4) + √ (J − 4)3/2
4
3 2
1
(1 +
2
(50)
Coming back to the behaviour at a given value of J, one can get an approximation
of the ’-’ solution. The fixed point equation for η for given values of J and h, is
η
)
η = H(η) ≡ 1/ 1 + exp (1 − h − 2Jη +
1−η
(51)
The ’-’ solution corresponding to a small value of η can be found by iterating
η(k + 1) = H(η(k)) starting with η(0) = 0, and η(k) is an increasing sequence
of approximations of η− . The lowest non trivial order is then given by
0
η−
= H(0) = 1/ [1 + exp (1 − h)]
20
(52)
which is indeed small for h strongly negative. At the next order
0
1
η−
= H(η−
)
(53)
0 as the small parameter, the expansion of η 1 gives
Taking η−
−
1
0
0
η−
= η−
(1 + (2J − 1)η−
)
(54)
and this gives the corresponding approximations for the price and the profit,
0
p1− = 1 − (J − 1)η−
0
0 2
Π1− = η−
+ J(η−
) .
(55)
0
It is clear from the above equation that the dependency on J is weak since η −
is small, in agreement with the exact behaviour computed numerically, shown on
figure 2.
Now we consider the neighbourhood of (J, h 0+ (J)), that is h = h0+ (J) + . De0 at h = h0 (J), η 0 (h) = η (J) + η (J)(1 − η (J)).
noting by η0 (J) the value of η−
0
0
0
+
−
Taking this expression for computing the profit of the ’-’ solution, and writing that
at the transition the two solutions ’+’ and ’-’ give the same profit, one gets the
following approximation for the value c (J) of at the transition (hence the value
of h at the transition, hc (J) = h0+ (J) + c (J)):
c (J) =
η0 (J)
0 (J)
η+
(56)
0 (J) is given by equation (47). It is this curve h (J) = h0 (J) + (J) which
where η+
c
c
+
is plotted in figure 3 for J > JB = 4.
21
Figure 5:
Functions y = G(h, J, η), y = η, price p(η) and profit Π(η) (+−). The intersects of y = G(h, J, η)
with y = η give the extrema of the profit; the (possibly local) maxima are those for which dΠ/dη > 0. Shown
here are marginal cases where for one solution dΠ/dη = 0, that is y = G(h, J, η) is tangent to y = η (points
on the lower or upper curves of the phase diagram, figure 3).
monopolist’s price
1
0.8
0.8
eta, beta p, profit(+)
eta, beta p, profit(+)
monopolist’s price
1
0.6
0.4
0.2
0.6
0.4
0.2
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
eta
(a) J = 27/8 h = −3/4 − log(2)
0.6
0.7
0.8
0.9
0.8
0.9
(b) J = 4 h = −2
eta+ : tangeance
eta+ : p=0
1
1
0.8
0.8
eta, beta p, profit(+)
eta, beta p, profit(+)
0.5
eta
0.6
0.4
0.2
0.6
0.4
0.2
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
eta
0.5
0.6
0.7
eta
(c1) Point on the lower curve, η < 1/2
(c2) Point on the lower curve, η > 1/2
(d1) Point on the upper curve, J > 4
(d2) Point on the upper curve, J < 4
disparition de eta- par tangeance
1
0.8
0.8
eta, beta p, profit(+)
eta, beta p, profit(+)
disparition de eta- par tangeance
1
0.6
0.4
0.2
0.6
0.4
0.2
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
eta
0.1
0.2
0.3
0.4
0.5
eta
22
0.6
0.7
0.8
0.9